THE ROLE OF THE SHAFT IN THE GOLF SWING

J. Bmmechanics Vol. 25, No. 9, pp. 975-983. 1992. 

Printed in Great Britain 

THE ROLE OF THE SHAFT IN THE GOLF SWING 

RONALD D. MILNE* and JOHN P. DAVIS? 

Departments of *Engineering Mathematics and JCivil Engineering, University of Bristol, 

Bristol BS8 ITR, U.K. 

CQZl-9290/92 %S.oO+.OO 

Pergamon Press Ltd 

Abstract--Current marketing of golf clubs places great emphasis on the importance of the correct choice of 

shaft in relation to the golfer. The design of shafts is based on a body of received wisdom for which there 

appears to be little in the way of hard evidence, either of a theoretical or experimental nature. In this paper 

the behaviour of the shaft in the golf swing is investigated using a suitable dynamic computer simulation 

and by making direct strain gauge measurements on the shaft during actual golf swings. The conclusion is, 

contrary to popular belief, that shaft bending flexibility plays a minor dynamic role in the golf swing and 

that the conventional tests associated with shaft specification are peculiarly inappropriate to the swing 

dynamics; other tests are proposed. A concomitant conclusion is that it should be difficult for the golfer to 

actually identify shaft flexibility. It is found that if golfers are asked to hit golf balls with sets of clubs having 

different shafts but identical swingweights the success rate in identifying the shaft is surprisingly low. 

INTRODUCTION 

‘We challenge anyone to prove the shaft is not the 

most important component of a golf club’. This bold 

statement accompanied a recent advertisement for 

golf clubs in a popular golf magazine and is typical of 

the emphasis that is currently being placed on the role 

of the shaft in golf club design. Shafts are currently 

being manufactured to closer tolerances than ever 

before and in a wider range. What is in doubt is the 

basis of the design philosophy. Roughly speaking, the 

received wisdom dictates that the more flexible shaft is 

better suited to the weaker player, while the profes- 

sional or good amateur should use a stiffer shaft. 

Considered as an empirical result arising from the 

long development of golf, it must be taken seriously; 

however, within the context of the dynamics of the 

golf swing, it is not an obvious conclusion. A typical 

explanation (Tolhurst, 1989) of the effect of shaft 

flexibility runs along the following lines (for those 

unfamiliar with golf, a glossary of terms is given in 

Appendix 1). The shaft is initially bent backwards in 

the early part of the downswing and then recovers just 

before impact. If the shaft is too flexible for the golfer, 

it springs forward too far, thereby increasing the 

effective loft and closing the clubface; if too stiff, it is 

still bent backwards, the loft is decreased and the face 

is open. In this sense the shaft should be matched to 

the golfer’s swing speed and hand action. Claims are 

sometimes made for increased clubhead speed at im- 

pact, assuming that the golfer’s timing can utilise the 

springing back of the shaft. The link between change 

in loft and closing or opening of the clubface is due to 

the clubhead being attached to the shaft at the lie 

angle (Appendix 1). The actual extent to which the loft 

is changed by shaft bending can be controlled by the 

location of the so-called kick-point which relates to 

the shape of the bent shaft at impact; a shaft with a 

Receiued in final form 8 January 1992. 

915 

low kick-point shows greater curvature near the tip 

region than one with a high kick-point. 

Shaft design is currently verified by three basic tests. 

In the first the shaft is clamped at its butt over a 

standard length and then loaded transversely by a 

weight near the tip; the tip deflection yields a measure 

of the shaft bending stiffness, and the location of the 

maximum deviation of a chord joining the butt and 

tip from the deflection curve yields the kick-point. The 

second test measures the tip rotation due to an 

applied tip torque, so yielding the torsional stiffness. 

The third test measures the fundamental transverse- 

vibration frequency of the clamped shaft carrying a tip 

mass equivalent to a clubhead. 

The purpose of this study was to gain an under- 

standing of the flexing behaviour of the shaft during 

the swing by constructing a suitable dynamic model 

and to test the validity of the model experimentally. 

Shaft flex may affect the feel of a golf club during the 

swing; this difficult area warrants a full investigation 

on its own but results of a rather limited study are 

reported here since they add some weight to the 

conclusions. 

Computer simulation 

METHODS 

Flexibility of the shaft is manifest in bending and 

twisting. The former is the subject of careful design, so 

that desired characteristics are obtained, whereas the 

latter is seen as a necessary evil, which must be 

minimised. In this study emphasis is on bending flex- 

ibility; consequently, the dynamic model is a develop- 

ment of the well-established plane two-link model of 

the golf swing (Budney and Bellow, 1979; Daish, 

1972). This consists of an arm link driven by a shoulder 

torque and a club link driven by a wrist torque; the new 

element added to the established model is the bending 

flexibility of the club link. Although the bodily mo- 

tions of a golfer are quite complicated, numerous

916 R. D. MILNE and J. P. DAVIS 

photographic and kinematic studies have shown that 

the downswing is, indeed, executed more or less in a 

plane. One aspect of the real three-dimensional swing, 

which is imported into the essentially two-dimen- 

sional or plane model, derives from the fact that the 

centre of mass of the clubhead does not lie on the 

shaft. Due to the large centrifugal forces involved in 

the swing, this has an important effect on shaft bend- 

ing and, consequently, for this reason, it is necessary 

to allow for rotation of the clubhead in the plane of 

the swing, although this rotation has no direct effect 

on the momentum balances or impact mechanics in 

the plane. A description of the full equations of mo- 

tion together with details of the computation are set 

out in Appendix 2. 

One integral feature of the model is crucial and 

needs emphasis. The shaft has a bending stiffness, 

which is measured in the standard cantilever bending 

test already referred to. However, during the swing, 

particularly near impact when the centrifugal forces 

are large, the shaft also acquires an additional bend- 

ing stiffness due to the fact that the shaft is under 

considerable tension. In a full drive or similar- 

distance shot this bending stiffness due to tension is, in 

the 40 ms or so before impact, large enough to be 

comparable to the conventional bending stiffness. 

Strain gauge measurements 

Three golfers, labelled A, B and C, with handicaps 

of 11, 5 and 2, respectively, were used in these 

experiments; golfers B and C showed a high degree of 

consistency in repeating their swings. The club used 

for the tests was a metal driver with R shaft and 

having 10.5” of loft. Groups of foil strain gauges 

(Showa, 2 mm) were bonded to the shaft at three 

stations to measure two bending moments at each 

station, parallel and normal to the clubface. Tension 

and torque were also measured at one station. The 

gauges were calibrated directly for these moments and 

forces by appropriate static loadings of the shaft. The 

strain gauge amplifiers (Fylde mini-bal and mini-amp 

systems) transmitted data via an analogue/digital con- 

vertor (Barr-Brown PCI-20089) directly onto a com- 

puter (Toshiba 3100SX) controlled by data acquisi- 

tion software (Labtech Notebook); data was stored in 

a form suitable for subsequent analysis using the 

spreadsheet Lotus l-2-3. The typical run extended 

over 10 s beginning at address and ending at follow- 

through. Sampling was at the rate of 200 readings per 

second; at this sampling rate it was not possible to 

read all the gauges simultaneously and successive 

swings were used. Graphical displays of data could be 

obtained on the Toshiba screen immediately after a 

run in order to check for failure or inconsistency. 

Initial analysis located the region of interest from just 

before top of swing to shortly after impact by refer- 

ence to the obvious point of impact with the ball. The 

swing was also observed by a single high-speed video 

system running at a rate of 200 frames per second and 

shutter speed of l/10,000 of a second placed facing the 

golfer; the video system was an NAC Model HSV400 

with associated X Y-coordinator Model V-78-E 

coupled to an IBM-compatible PC using the motion 

analysis package MOVIAS (version 3). A video re- 

cording was also made along the line-of-flight direc- 

tion to establish the angle of the golfer’s swing plane. 

The video recordings allowed the gauge data to be 

correlated with swing position through the common 

time of impact and also allowed the construction of 

stick diagrams for the swing and the estimation of 

clubhead velocity and acceleration. The most import- 

ant information obtained from the recordings gave 

the orientation of the clubhead relative to the plane of 

the swing. This was needed to transform the parallel- 

to-clubface and normal-to-clubface bending-moment 

components to in-swing-plane and out-of-swing 

plane components since only the in-swing-plane com- 

ponents are relevant to the simulation. This aspect of 

the experimental method was the least satisfactory. 

The clubface was marked and the video image was 

clear; nevertheless, an accurate measurement of the 

orientation was difficult. Typically, the clubhead 

rotates little during the downswing until within about 

50 ms of impact when it rotates through approx- 

imately a right-angle to square the clubface at impact. 

A simple smoothing routine was used to obtain both 

in-plane and out-of-plane bending moments. Out-of- 

plane bending moments were very small except near 

impact, where the values were consistent with out-of- 

plane bending of the shaft due to centre-of-mass offset: 

the behaviour of the resolved bending moments serves 

to give confidence in the plane-swing model. Were 

experiments of this type to be repeated, a more accur- 

ate and automatic way of measuring orientation, 

perhaps using laser marker techniques, would be 

advisable. Questions of cost precluded such an ap- 

proach in these experiments. 

RESULTS 

A set of results is shown in full for golfer B only; 

results for golfers A and C naturally differ in detail but 

not enough to alter the overall picture as far as the 

shaft behaviour is concerned. A typical set of results 

for a swing simulation is shown in Fig. 1. Clubhead 

speed at impact estimated from the video via the 

MOVIAS program was 41.4 ms-‘. The initiation 

time for the swing simulation was chosen as 400 ms 

before impact, at which point in the backswing the 

shaft is straight and the wrist torque is just beginning 

to reverse: angular velocities at initiation were estim- 

ated from the video recording. The diagrams have 

been extracted from a continuous screen display to 

illustrate the main features. The club used for the 

simulation is a driver; a long iron club would show a 

very similar swing except that the bending forward of 

the shaft before impact would be much reduced. The 

stick diagrams in Fig. 1 show the shaft deflection to 

scale, and it is difficult to perceive detail; so, the shaft

Time before impact 68 ms Time before impact 39ms 

Time before impilct l9ms 

Time after impact IOms Time after impact 24ms 

Role of shaft 911 

Sha ft.c+fkct+n 

wn’f’ed ’ bmes 

Driver-Iaft 10.5&9 lnyuct &city 41.6mIs Drier-loft Kl.5dcp Imct velocity Ll.Cm/s 

Clubface aqk ll.Bdq Ball vebcity 5X6m/s Ctubfact aqk 11.8de9 Ball velocity 5Z6m/s 

deflection is also shown magnified five times. These 

bending shapes are plotted in such a way that they 

have the property of being orthogonal (with respect to 

the club mass distribution) to a rigid-body rotation 

about the wrist hinge (Appendix 2). Other outputs 

from the simulation, shown in Fig. 2, are clubhead 

speed, clubhead deflection measured relative to a tan- 

gent at the butt end and hand force. The hand force is 

the internal vector force at the wrist hinge acting in 

opposite directions on the arm and club links; in 

Fig. 2 the components of this force are shown resolved 

along and normal to the instantaneous directions of 

both arm and club links. Figure 2 also shows the 

tension measured in the shaft at its mid-point: meas- 

ured mid-point torque is not shown, as it did not 

exceed 0.5 Nm until impact. In Fig. 3 are shown the 

simulated and measured in-swing-plane bending mo- 

ments at three stations along the shaft-at 10% (top), 

Fig. 1. Swing simulation showing shaft deflection. 

r-i!- 

1 

50% (middle) and 90% (bottom) of shaft length from 

the wrist hinge. 

Shoulder and wrist torques for the simulation ap- 

propriate to golfer B were synthesised from the strain 

gauge data in the following way. The shoulder torque 

can be estimated from the measured shaft tension by 

utilising the fact that the force exerted on the shaft by 

the hands is almost wholly axial to the shaft through- 

out the swing, the tangential component of the hand 

force being at most about 10% of the total; hence, 

shaft tension is little different from the total hand 

force, a result which can readily be demonstrated 

theoretically. By using a point in the swing at which 

the angular acceleration of the arm link is zero (about 

70 ms before impact) the inertia of the arms/torso is 

eliminated from the estimate. The agreement between 

axial (club) hand force and mid-shaft tension (Fig. 2) is 

very satisfactory when it is noted that the tension at


tim ms time ms 

- Axial 

---- Kangenttil 

‘-‘- shaft tension 

Fig. 2. Swing simulation outputs and measured mid-shaft 

tension. 

10. ______- ----__ ___ 

--._ 

g-1: 1 -f----- % t 

-10 

\_I-- 

0 100 ZW 300 100 SW 

Time ins 

Fig. 3. Simulated and measured shaft bending moments. 

mid-shaft is approximately 90% of the tension at the 

top. The wrist torque can obviously be estimated from 

the measured bending moment at the top station; the 

nearer this station is to the wrist hinge the more 

accurate the estimate. Based on this evidence the 

shoulder torque was applied as a ramp function with 

rise time 110 ms and maximum amplitude 83 N m and 

the wrist torque was applied as a ramp function with 

rise time 275 ms and maximum amplitude 25 Nm. 

These simplified torques were not intended to match 

the golfer’s inputs exactly. The inputs for golfers A 

and C were qualitatively similar to those for golfer B 

and the level of agreement between simulation and 

measurement much the same; golfer C was a very long 

hitter generating a clubhead speed of 48 m s-l. 

INTERPRETATION OF RESULTS 

For the purpose of describing the shaft behaviour 

the swing may be divided into three phases. In the first 

phase, which extends from about 230 ms before im- 

pact (top of the swing) to about 130 ms before impact, 

the application of the wrist torque bends the shaft 

backwards, the maximum deflection occurring near 

the end of this phase. In the second phase, which 

consists of the last 130 ms or so to impact, an unfold- 

ing of the system due to momentum transfer takes 

place. The shaft behaves in a quasi-steady manner, 

gradually straightening and then bending forward due 

to the anticlockwise centrifugal torque applied at the 

lower end of the shaft by the offset of the centre of 

mass of the clubhead behind the shaft centre-line. This 

torque is closely represented by the bending moment 

at the bottom station (Fig. 3); near impact the effective 

clubhead weight is some 150 times its actual weight, 

so the magnitude of the torque applied is some 3 N m 

for each centimetre of centre-of-mass offset. The main 

effect is to give the club an additional dynamic loft at 

impact with a corresponding closing of the clubface. 

The nett clubface angle at impact with the ball is 

recorded by the simulation (Fig. 1). It is important to 

note that the position of the centre of mass of the 

clubhead relative to the hands is along the local 

centrifugal vector, so that if the shaft is bent forward 

then the club butt-to-arm angle is correspondingly 

increased and vice versa (see also Fig. 4). In phase 

three the effect of impact on the shaft is to excite a few 

cycles of bending vibration at a frequency of about 

30 Hz. Some of the energy at impact is spent in 

exciting the shaft; for example, given the same set of 

input torques, the simulation shows that a flexible 

shaft produces an initial ball velocity which is about 

4% lower than that produced by an equivalent ideal- 

ised ‘rigid’ shaft. 

The computer simulation necessarily incorporates 

a model of the dynamic characteristic or impedance of 

the wrist joint, and this is an area where information is 

lacking: experimental work such as that carried out by 

Brown et al. (1982) on other human joints is badly 

needed. When gripped by a golfer, the club is very far 

from being clamped, as in the standard bending and 

vibration tests. The stiffness or the in-phase compon- 

ent of the wrist impedance torque is probably quite 

small when, near impact, the joint is relaxed and 

rapidly developing. The damping or the out-of-phase 

component is large enough near impact to actually 

apply a retarding torque. The measured top station 

bending moment in Fig. 3 clearly shows the decrease 

of the wrist torque approaching impact as the rota- 

tion speed of the joint rapidly increases. This behavi- 

our was observed for all the golfers tested and has 

been observed by others (Budney and Bellow, 1979); 

details of the impedance used in the model are given in 

Appendix 2. By varying the impedance-damping level 

and using the same demand inputs, the simulator can 

arrive at impact with wrist torques which vary from 

decelerating (the usual case.) through neutral to accel- 

erating. The results from three simulations of this 

type, based on the swing of golfer B but taken over a 

rather extreme range for emphasis, are shown in 

Fig. 4. Note that the clubface angle at impact is the

-_ 


r 

SWI defkction 

1982; Cochran and Stobbs, 1969) and confirm an 

maaniti& 5 times 

lime betore impact 4 ms 

earlier description of shaft behaviour which was based 

lIecekr&g wrist torque 

solely on well-founded physical reasoning (Williams, 

at inplct I- mNml 

1969). 

! 

DISCUSSION 

The main aim of the work reported here was to 

elucidate, in a qualitative way, the dynamic flexing 

1, -5 0 cm 

behaviour of the golf shaft during the swing. In the 

Drim-loft m.5* Impct vrbcityQ.9m/s 

simulation the precise torques generated by the 

Clubface u?pk B8dcg Bali nkcity 566 m/s 

‘golfer’ are open to question but the inertial and 

he before itquct 3ms 

MC&a/ wrist ivrque 

at impact (0 Nm) 

1, 

imposed forces and torques applied to the shaft itself, 

L-r 

as verified by the strain gauge measurements, are 

undoubtedly typical of the golf swing and, hence, the 

level of confidence that may be placed in the simulator 

is high. The simulator could be used as a design tool 

to explore in detail the interaction between a range of 

shafts and ‘golfers’ as represented by their input 

torques. No attempt is made to do this here, but it is 

worth remarking that widely different shoulder 

-5 0 cm 5 torques lead to similar simulated swings provided the 

Driver-loft 10.5dq Impct v&city 122m/s 

same amount of work is done: this is not a surprising 

Ctubfaa an9k 12&q Ball nlocity 58.4 m/s 

result if one remembers that the swing is the outcome 

Time be/on impact 2ms 

of a double time integration and is reflected in the 

observation that, on the golf course, very different 

looking golf swings can produce similar results. 

The burden of the conclusions arising from the 

study is that shaft flexibility does not play an important 

dynamic role in the swing and the principal effect 

&her-btt M5de9 1-t bvtocity 43.5m/s 

of flexibility, namely, change in the effective loft at 

impact, could be estimated from static considerations. 

A simple test which would serve this purpose is illustrated 

in Fig. 5(a); this simulates the behaviour of the 

shaft of a wooden club just before impact when condi- 

Cl&face qle 13.Ldq BaX v&city 60 1 m/s 

tions are quasi-steady and the shaft is subject to a 

Fig. 4. Effect of wrist torque on shaft shape at impact. large axial load. The outcome of the test is a direct 

measure of the change in the effective loft (per unit 

centre-of-headmass offset) which this shaft would give 

result of two opposing effects, the change in butt-to- 

arm angle. being counteracted by the change in rota- 

tion at the shaft tip. The impact velocity is increased 

only by 2.6 m s- ‘, which is consistent with the fact 

that, for the whole swing to impact, approximately 

92% of the work done comes from the shoulder 

torque (via the torso and legs) and only 8% from the 

wrist torque. 

The clamped bending frequency of a driver is typi- 

cally of the order of 4.5 Hz, whereas the bending 

frequency observed for a club freely hinged near its 

butt end as illustrated in Fig. 5(b) is of the order of 

25 Hz. The bending frequency for a club, freely rota- 

ting about a hinge near its butt end and having a 

speed of rotation which gives the head a speed of 

40 m s- I, is about 32 Hz as observed in the simulation 

after impact; in this situation the frequency is only 

weakly dependent on the shaft bending stiffness. The 

three phases described above are all observable in 

high-speed photographs of golfers’ swings (Maltby, 

(b) m 

Changeotlottdwtooftaotofcmtmof 

- cd clubhud lrom sh#t cantm IIM 

ShCftICplVOtCdCtbldtCndlOCdWIC 

rppScd vlc rigId bm&ct ct lip 

Appllcalon 01 wlcl torque un bc 

*mummd by applybIg mothnr load via 

� rtgld bmcket atUch#d at butt 

Lcwca ncxuml Vlbmtlon mode 01 Divomd 

ShM with bndrmu 

Fig. 5. Proposed shaft tests.

980 

and is related to the player’s clubhead speed at impact 

if the load W is given a value appropriate to the 

centrifugal force on the head; it automatically takes 

account of the change in club butt-to-arm angle al- 

ready referred to earlier. Application of a torque at the 

butt end can simulate the golfer’s wrist torque at 

impact. This test is reflected in the common practice 

amongst golfers of resting the toe of a club on the 

ground and then pushing the butt firmly downwards! 

Clearly, another test which should be included in 

shaft testing is the measurement of the fundamental 

bending-vibration frequency of a shaft-plus-head sus- 

pended freely from a pivot as illustrated in Fig. 5(b). 

To demonstrate the effect of an increase in the bend- 

ing stiffness due to centrifugal force, approximate 

vibration frequencies are shown for a rotating shaft- 

plus-head and for a rotating chain-plus-head, the 

chain having the same mass distribution as the shaft 

but, of course, zero conventional bending stiffness. 

If shaft flexibility is not dynamically significant then 

why is it apparently so important to the golfer? To 

begin to answer this question in a very preliminary 

way a series of trials was conducted with a small 

group of golfers consisting of four amateurs, with 

handicaps ranging from 13 to 5, and four profes- 

sionals. Three sets of clubs were specially prepared, 

each set having identical swing weights but fitted with 

R, S and X shafts, two sets of five-irons (with solid 

back and peripherally weighted heads) and a set of 

drivers. The decals on the shafts were covered by a 

code letter and the participants were instructed not to 

bend the shaft or to waggle the club but simply to hit 

golf balls. The results were somewhat surprising. The 

amateurs were unable to distinguish one shaft from 

another either for woods or irons; typically, if pressed 

as to which shafts they preferred, they would chose the 

R and X over the S or, if asked for an order of 

flexibility, would give an almost random selection. 

Striking of the ball did not seem to be greatly affected, 

although no measurements were made of ball traject- 

ory. The professionals were not much better at dis- 

criminating between the irons but were quite success- 

ful with the woods. However, even here success was 

not complete, although the R shaft could generally be 

distinguished from the S and X shafts. These very 

simple and restricted tests do suggest that more exten- 

sive and significant work should be undertaken in this 

area. Recently, Pelz (1990) has conducted ‘blind’ golfer 

trials to compare the performance of steel and graph- 

ite shafts of differing flexibilities and Van Gheluwe et 

al. (1990) have shown that the kinematics of a golfer’s 

swing appears to be independent of the shaft material. 

CONCLUSIONS 

Do the results of this study bear out the accepted 

explanation of the effect of shaft flexibility given in the 

Introduction? The study confirms that flexibility will 

produce a change of loft (and close clubface) at im- 

pact but shows that these are really quasi-static effects 

R. D. MILNE and J. P. DAVIS 

and not dependent on a dynamic ‘springing back’ of 

the shaft from its bent position at the start of the 

swing. It is difficult to give any credence to the sugges- 

tion that the shaft can actually be bent backwards at 

impact and no photographic evidence exists to sup- 

port this contention. Kick-point should be replaced 

by a measure of shaft-tip rotation due to an offset 

axial load. Nor does springing back or ‘whipping’ of 

the shaft play a dynamic role in the all-important pre- 

impact area, although the ‘feel’ of the violent flexing of 

the shaft after impact may give the golfer the impres- 

sion that it does. Indeed, clubhead speed at impact is 

virtually unchanged if, using the simulator, the shaft is 

replaced by a heavy chain about 60 ms before impact. 

Perhaps the major source of misunderstanding about 

the role of the shaft is associated with the idea that the 

club is ‘gripped’ by the golfer, resembling the condi- 

tion of the built-in butt end of the cantilever test, 

combined with a lack of appreciation of the magni- 

tude and speed of response of any forces and moments 

applied by the player, which could significantly affect 

the course of the swing in the pre-impact area. 

The implications of this study for club design are 

somewhat problematical. It does appear that it may 

not be necessary to provide a range of shafts for iron 

clubs and possibly not even for wooden clubs. It is 

possible to argue that the ‘feel’ imparted to a club 

through shaft flexibility may actually affect the torque 

inputs of the player but the golfer tests gave no 

indication of this: there is room here for further study. 

If one takes the view that shaft flexibility is not of 

importance dynamically, it then represents an un- 

wanted swing variable and one is led to adopt a stiff 

shaft, a conclusion also reached by Pelz (1990). The 

effective loft that a golfer may have enjoyed with a 

fiexible shaft could then be reinstated (without the 

associated change in clubface angle) by an appropri- 

ate adjustment of nominal club loft. Do current stiff 

shafts represent an upper limit or could stiffer shafts 

be used? If the latter, then it would be possible to use a 

light, thin-walled shaft tapering to about twice the tip 

diameter of the currently available shafts coupled with 

a return to hosel-in-shaft fabrication; such a shaft 

would have considerably higher torsional stiffness 

than the currently available shafts. 

Finally, the point should be made that the fixity 

conditions of a shaft held in a golf machine are very 

different from those for a golfer and, therefore, the 

shaft performance data derived from this source do 

not necessarily apply to the ‘real’ golf swing. 

Acknowledgements-The authors would like to express their 

gratitude to Dunlop Slazenger International Ltd for finan- 

cial aid and for supplying golf clubs. 

REFERENCES 

Brown, T. I, H., Rack, P. M. H. and Ross, H. F. (1982) Forces 

generated at the thumb interphalangeal joint during im- 

posed sinusoidal movements. .I. Physici. 332, 69-85. 

Budney, D. R. and Bellow, D. G. (1979) Kinetic analysis of a 

golf swing. Res. Quart. Exercise Sport 50, 171-179.

Cochran, A. and Stobbs, J. (1969) The Search for the Perfect 

Swing. Heinemann, London, England. 

Daish, C. B. (1972) The Physics of Ball Games-Part II. 

English Universities Press, London, England. 

Maltby, R. (1982) Golf Club Design, Fitting, Alteration and 

Repair. Ralph Maltby Enterprises, Newark, NJ. 

Pelz, D. (1990) A simple scientific shaft test: steel versus 

graphite. In Proceedings of the First World Scientific Con- 

gress of Golf (Edited by Cochran, A. J.), pp. 264269. 

Chapman & Hall, London, England. 

Tolhurst, D. (1989) What you need to know before you buy. 

Golf Monthly Equipment Supplement, April 1989 issue. 

Van Gheluwe, B., Deporte, K. and Ballegeer, K. (1990) The 

influence of the use of graphite shafts on golf performance 

and swing kinematics. In Proceedings of the First World 

Scientific Congress of Golf (Edited by Cochran, A. J.), pp. 

258-263. Chapman & Hill, London, England. 

Williams, D. (1969) The Science of the Golf Swing. Pelham 

Books, London, England. 

APPENDIX 1 

GLOSSARY OF GOLFING TERMS 

The golf club consists of three parts: the head, the shaft and 

the grip. The head is either a fairly narrow metal blade (iron) 

or a more extended, bulbous shape with a flattish front face 

(wood) made traditionally of wood but now also made in 

metal. The loft of the club is the angle which the front face of 

the head (clubface) makes with the vertical. The shaft is 

attached to the head by the hose1 or neck, which is a cylindri- 

cal extension of the head situated near one end of the 

clubface. The hose1 is angled to the head in such a way that 

when the base of the head is resting flat on the ground the 

shaft is at an angle to the horizontal called the lie angle. The 

centre of mass of an iron club is situated about halfway along 

the blade while the centre of mass of a wooden club is offset 

from the neck roughly along a line at 45” to the clubface. The 

shaft is either a steel tube or a solid composite tapering down 

from the grip end (butt) to the tip. The distribution of the 

taper controls the bending and torsional properties of the 

shaft. Shafts are commonly manufactured in three main 

categories R (regular), S (stiff) and X (extra stiff): within each 

category different weights and kick-point positions are 

offered. Typically, the ratio of clamped transverse vibration 

frequencies for R, S and X shafts is 1.00: 1.04: 1.08. The grip is 

simply a rubber tube which is glued to the butt of the shaft. 

At impact the clubface is said to be open ifit is pointing to 

the right of the target line and closed if it is pointing to the 

left: the efictioe lof is the actual angle of the clubface to the 

vertical at impact as opposed to the nominal loft of the club. 

Bending of the shaft in the swing plane near impact not only 

changes the loft but also opens or closes the clubface; for 

example, for a driver with lie angle of 55” the clubface will 

open (close) by an angle which is 70% of the change in the 

loft. 

The swing plane is the imaginary plane in which the player 

swings the clubhead and will generally be inclined somewhat 

steeper than the lie angle of the club. 

Due to the arrangement of bones and joints in the lower 

arms and wrists, the golfer is forced to roll or rotate his 

hands during the golf swing if he is to allow the wrist hinge to 

move freely: the face of the club lies roughly parallel to the 

swing plane at the top of the swing and is then rotated so as 

to be at right-angles to the swing plane at impact. 

APPENDIX 2 

THE MATHEMATICAL MODEL 

Equations of motion 

Figure Al illustrates the applied torques and rotation 

angles for the dynamic model. The relevant equations of 


Fig. Al. Applied torques and rotation 

dynamic model. 

angles for the 

motion, listed in the following, consist of 

(a) equations (1) and (2) representing the rate of change of 

angular momentum (including terms due to shaft flexibility) 

about the upper (fixed) pivot and about the wrist hinge, 

respectively, and 

(b) a variational equation (principle of virtual workj for 

shaft bending based on the small-deflection theory for slen- 

der beams including the effect on bending of tension in the 

shaft due to centrifugal forces. This integro-differential equa- 

tion is replaced by a number of ordinary differential equa- 

tions in time by representing the shaft bending displacement 

as a series of shape functions with time-dependent ampli- 

tudes (the Galerkin method) leading to equations (3), , 

(n+2). 

Equation (1): 

(I,+m,l~)ti+ m,l,7,cos(+B) 

( 

-(i, EiCi+C)sinCO-B))$ 

+ i E,cos(~-0)- 

i=1 

-(m.l,,csin(4-8)+ 

-2 i E,&sin(4-0)$ 

i=1 

+gsin0(l,m,+m,L)+ T,- T,=O, 

Equation (2): 

m,l,l;cos(#-0)- 

+(~, E,c,+c)cos))+T,rO. 

Equations (Z+i), i=l, . . . , n 

E,(sin(&e@+cos(4-e)i)+ i k,(T,+&) 

,=1 

+(cos(4-0)@-sin(+e)l) i; G,,rj+H,C 

j=l


- T,s;(o)/l,=o, 

G,,=f s ’ Ca(u)+mbls;(u)s;(u)du, 

+y, 

s 0 

* i 

s 

M,,=Pok P,(~W~,W du+wi(lb,W, 

0 

s 

1 

K,‘= [rk,(u)+m&;(u)s;(u)du, c=l,m,E, 

0 

ei(u)s;(u)s;(u)du, 

T,=Tt+imp@-4, e-4); 

dot denotes differentiation with respect to time, t, 

whereas prime denotes differentiation with respect to a, 

Z( 4) = distance of c of m of clubhead from shaft in 

swing plane, 

ei(u) = bending stiffness ratio of shaft, ei(0) = 1, 

EI = reference bending stiffness of shaft (at u = 0), 

&j=gcos(swp) 

(J = gravitational acceleration, 

{ swp = angle of swing plane to vertical, 

Jr = moment of inertia of arms/torso about upper 

pivot, 

I, = moment of inertia of club about wrist pivot, 

imp( ) = wrist impedance function, 

I, =length of arm link, 

1, = length of club link, 

L = distance of c of m of arm link from upper pivot, 

t = distance of c of m of club link from wrist pivot, 

m, = mass of club, 

m, =mass of clubhead, 

a = number of shape junctions, 

s,(u) = ith shape function, 

t = time, 

T, = torque at upper pivot (‘shoulder torque’), 

T, = torque at wrist pivot (%rist torque’), 

T”, =‘demand wrist torque’, 

a = material,damping constant for shaft, 

g(t)=angle of arm link to vertical, 

C(t) = amplitude of ith shape function, 

p,(u) =mass density ratio of shaft per unit length, 

p(O)= 1, 

p. = reference mass density of shaft (at u = 0), 

u = dimensionless distance along shaft from wrist 

pivot, 

$(t) = angle of club link axis to vertical (Fig. Al). 

First-orderform of equations of motion 

where 

f,=z, 

“+Z 

,TI m,‘({x~})i’=Ft({x~},Izt)) 

) 

i,k=l, . . ..n+2. 

{xsl=(@, 4, L 52,. . . . 9 5.1. 

In this form the equations of motion are integrated forward 

in time using a fourth-order Runge-Kutta scheme. 

Momentum balance equations at impact 

During forward integration of the equations of motion, 

impact with the ball is detected by an interpolation of the 

clubhead path between time steps: the program enters a 

routine which solves the following algebraic equations representing 

conservation of momentum of the whole system 

and then reenters the integration program with a new set of 

initial values, the first time step being adjusted to lock onto 

the pre-impact time markers; 

“C7. 

C m,,({x~“‘})(zjf’-zy’)=m,uu,, 

‘==I 

i,k=l,...., n+2, 

n+2 

C u,(z~f)+ez~i))=u i 

i=1 

where 

{Ui} =(l,cosf+‘, l,cosfjP~), s,(l)cosf$“m), 

..., s ” (1)cosQ;““‘) 

$m) = (j+m) + y, $im) = &0 + ?, 

m, = mass of golf ball, 

v = initial ball velocity, 

e = coefficient of restitution, 

y =initial angle of ball flight path above horizontal, 

(im)*(i)s(‘)denote impact, initial and 6nal values. 

Shape functions-pre-processing program 

The n shape functions are defined by sr(u)=c;L: +a’, 

where the coefficients c,, are chosen such that, for all 

i,j= 1, . ..,n, 

1 

1 

~(~)s,(~)~d~+~~,(l)=4 

s 0 

p(u)si(u)s’(u)du+mssi(l)s,(l)=O, i#j. 

I 0 

That is to say, with respect to the mass distribution of the 

club, the shape functions are mutually orthogonal and are 

also orthogonal to the rigid-body rotation u. As a result, the 

left-hand side matrix cm,,] has non-zero entries on the 

diagonal and in the first row and first column only. Given 

the club mass and bending stiffness distributions, a preprocessing 

program generates the shape functions recursively, 

then computes the stiffness matrix [k,,]. The program 

also calculates the fundamental pivoted vibration frequency 

of the club [Fig 5(b)]. It is found that three shape functions 

are sufficient to give good accuracy for the frequency calcu-


lation and to deal with the changes in shape of the shaft for the three golfers, force demand being entirely embodied 

which occur throughout the swing simulation. in 7’:. An alternative model using a kinematic demand on 

club/arm angle and embodying also a non-linear stiffness 

Wrist impedance function term weighted towards the limit of wrist rotation gives very 

The hypothetical impedance function consists solely of the 

damping term const. (O-c$)‘, with the same constant used 

similar results in the simulation.

THE ROLE OF THE SHAFT IN THE GOLF SWING

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